Totally real submanifolds of Kaehler product manifolds

Totally real submanifolds have been studied by many geometers in different ambient manifolds. The purpose of this note is to study totally real submanifolds in Kaehlerian product manifolds. We derive some integral formulas computing the Laplacian of the square of the second fundamental form and using these formulas we prove pinching theorems. In fact, we have generalized some results due to Yano and Kon [1,2] to the case when the ambient manifold is Kaehlerian product manifold.